Predicting Population Growth: Modeling with Projection Matrices - Mathematical Concepts and Methods in Modern Biology - page 215

The probability that a seed at time t 0 :
remains a seed is 0, therefore A 1 , 1 = 0,

becomes a seedling at time t 0 + 1 is 0.15, therefore A 2 , 1 = 0 . 15,

becomes a one-leaved plant at time t 0 + 1 is 0, therefore A 3 , 1 = 0,

becomes a two-leaved plant at time t 0 + 1 is 0, therefore A 4 , 1 = 0,

becomes a three-leaved plant at time t 0 + 1 is 0, therefore A 5 , 1 = 0,

becomes a four-leaved plant at time t 0 + 1 is 0, therefore A 6 , 1 = 0.

Note the probabilities do not sum to 1. This is because not all seeds (in fact only 15%)
survive to become seedlings in the following year.
From these probabilities, we know that the first column of the projection matrix is
as shown in Table 7.1 .
To generate the second column of the matrix, we must consider what happens to the
seedlings between the first year (time t 0 ) and the second year (time t 0 + 1). Seedlings

can either die or become plants with one leaf; they do not flower, so they do not make
seeds, and they cannot become bigger than a one-leaved plant in just a year. Therefore:
The probability that a seedling at time t 0 :
makes seeds at time t 0 +

1 is 0, therefore A 1 , 2 = 0,

stays a seedling at time t 0 + 1 is 0, therefore A 2 , 2 = 0,

becomes a one-leaved plant at time t 0 + 1 is 0.21, therefore A 3 , 2 = 0 . 21,

becomes a two-leaved plant at time t 0 + 1 is 0, therefore A 4 , 2 = 0,

becomes a three-leaved plant at time t 0 + 1 is 0, therefore A 5 , 2 = 0,

becomes a four-leaved plant at time t 0 + 1 is 0, therefore A 6 , 2 = 0.

From these probabilities, we know that the second column of the projection matrix
is as shown in Table 7.2 .
To generate the third column of the matrix, we consider what happens to plants
with one leaf between time t 0 and time t 0 +

1. We know from the diagram in Figure
7.1 that one-leaved plants have a probability of staying a one-leaved plant in the next

Table 7.1 The first column of the projection matrix for a population of Ameri-
can ginseng. The values in this column are the probability of transitioning from
the seed stage to another stage (data from [ 10 ]).
Stage at time t 0
Seed Seedling One- Two- Three- Four-

leaved leaved leaved leaved

plant plant plant plant

Stage Seed 0 ∗ ∗ ∗ ∗ ∗

at Seedling 0.15 ∗ ∗ ∗ ∗ ∗

time One-leaved plant 0 ∗ ∗ ∗ ∗ ∗

t 0 +1 Two-leaved plant 0 ∗ ∗ ∗ ∗ ∗

Three-leaved plant 0 ∗ ∗ ∗ ∗ ∗